Response Effects of Guitar Pickup Position and Width

J. Donald Tillman
1 July 2000, updated 17 October 2002

Introduction

When electromagnetic pickups were first applied to guitars the
goal was to make a louder instrument. I'm sure it didn't
take long to realize that the new electric guitar not only
sounded very different from an acoustic guitar, but the sound
was full of new musical potential. Also the timbre was
surprisingly dependant on the construction and placement of the
pickups as well as combinations of pickups.

A pickup placed near the neck has a deep rich sound while a pickup
placed near the bridge has a bright jangly sound. A pickup
placed in between those locations has a brassy sound. None
of these sound much like an acoustic guitar, or any other instrument.

On an acoustic guitar every component of the string vibration is
audible. Longitudinal waves, transverse waves, along any
axis, any direction, every harmonic; they all eventually find
their way to being a force on the bridge and thus a contributing
component of the sound of the instrument. On an electric
guitar only the displacement of the string at the pickup location
is sensed, and then only the displacement of transverse waves
along the axis of pickup sensitivity.

This concept of tapping off one specific aspect of a complex
mechanical interaction as an electrical voltage and presenting it
to an amplifier and speaker was a new process to use for a musical
instrument. Later developments along these lines gave us,
in roughly chronological order, the Hammond organ (electromagnetic
pickups on tonewheels), the electric bass (electromagnetic pickups
on strings), the Rhodes piano (electromagentic pickups on tines),
the Wurlitzer piano (electrostatic pickups on reeds), and the
Hohner Clavinet (electromagnetic pickups on strings). In all
these cases the electrical signal recovered from a mechanical
process sounds fundamentally different from any acoustic sound of
that same process.

Presented here is an analysis of some of the effects of pickup
position and width, including the derivation of frequency
response equations and plots based on those equations.
Also, since the Roland VG-8 Virtual Guitar System emulates
these very response curves I've included a short review of the
patents for that device.

I won't be covering the electrical characteristics of pickups,
the effects of multiple pickups, instrument body resonsances
or some of the other effects on the sound of an electric
instrument here. Another article,
Response Effects of Guitar Pickup Mixing,
examines some issues involved in mixing the signals from
multiple pickups.

Pickups and a vibrating string

Figure 1 is a drawing of a string such as you would find
on an electric guitar or electric bass. The nut is on the
left, the bridge is on the right and three pickups are positioned
in typical "neck", "middle" and "bridge" locations. The
drawing is to scale horizontally, but not vertically. The
string "scale length", or distance between the nut and the bridge,
is 25.5 inches. The neck, middle and bridge pickups are
positioned 6.375, 3.875 and 1.625 inches from the bridge
respectively. These values are directly modeled after the
Fender Stratocaster (see note below).

Figure 1. A vibrating open string with a scale
length of 25.5 inches and three pickups located at 6.375
inches (neck), 3.875 inches (middle) and 1.625 inches
(bridge).

The blue strips in the drawing designate the output level of
the vibrating string sensed at the pickup positions. Not
surprisingly, the output increases as the pickup position
approaches the center of the string.

I need to point out that I'm glossing over two effects that
cancel each other out. One is the fact that magnetic
pickups respond to the change in magnetic field, and thus the
velocity and not the displacement of the string. For a
given string and a given displacement, the pickup output will
grow proportionally as the frequency increases.

The other effect is that vibrating strings with a given energy
have a displacement that is proportionally less for higher
frequencies. You can observe this if you stare at the
guitar strings while playing an open string and its
harmonics.

So for a given string, the displacement decreases with
frequency, while the sensitivity of the pickup increases with
frequency. In these article I refer to the displacement
sensed by the pickup relative to the open string, letting the
pickup's sensitivity to velocity and the string displacement
dependance on frequency cancel each other out.

How about optical or electrostatic pickups? They are
sensitive to the string displacement instead of velocity and
will need a 6dB/octave high frequency boost to avoid being
bass-heavy.

Will lower-tuned strings show a lower output with magnetic
pickups? Yes, except that lower-tuned strings have more
moving mass, and thus more magnetic sensitivity, to
compensate. So it all works out.

Figure 1 is the simplest case; the string is vibrating at
the fundamental frequency on the "open" (or unfretted)
string. String vibrations also include sustained
harmonic frequencies as well as transient vibrations.
The sustained harmonic vibrations are of special interest
because they define the overtone series of the guitar's sound.

Figure 2 shows the same string vibrating at the fundamental and
second, third, fourth and fifth harmonics. The harmonics can
exist either as components of any played note, or forced out
individually by playing a note with a finger lightly pressed over
a node point on the string. (A "node" is a point where a
string has minimal motion. Besides the endpoint nodes at the
bridge and nut, harmonic vibration shows additional nodes in
between. An "antinode" is a point where the string has
maximum motion.)

Second harmonic. There is a node at the center of the
string and two antinodes located at 1/4 and 3/4 the length
of the string. In this drawing one of the antinodes is
located directly above the neck pickup.

Third harmonic. Two nodes, three antinodes. At this
higher pitch the output of the neck pickup is substantially
lower.

Fourth harmonic. This is an interesting case because there
is a node positioned exactly over the neck pickup and the pickup
will not have any output at this frequency. Many electric
guitar models have their neck pickup positioned one quarter of
the scale length from the bridge. If you have one of these
instruments handy, try comparing the sound of the open string
fourth harmonic on the various pickups.

Fifth harmonic. Here both the bridge and neck pickup
signals are more substantial. The strip is red here to
indicate that displacement is in the opposite direction, or
"out of phase", with respect to the string motion near the
bridge. If the two pickup signals are added together
there will be some cancellation of this harmonic.

Of course fretted notes are used for more often than open strings
on a guitar. For fretted notes the same harmonic vibration
patterns occur, subject to mechanical resonances of the instrument
and the physics of the string of course, but over a smaller
length of the string.

Figure 3. The fundamental, second, third and fourth
harmonics of a note played at the 12th fret.

Figure 3 is the same instrument as the previous figures, but with
a note played an octave up at the 12th fret instead of an open
string. As far as the pickup is concerned, the effects of
the nodes and antinodes are due to the wavelength on the string
and not whether the string vibration is a due to a fretted
fundamental or a harmonic.

Pickup response due to position

The task here is to derive an equation to describe the
variations of the frequency response of a pickup due to the
position of the pickup along the length of the string.
First we'll assume that the pickup senses the string motion at
exactly one point, then we'll move on from there.

My college physics text,
Halliday and Resnick,
includes two chapters that are very helpful; "Waves in Elastic
Media" and "Sound Waves". The book doesn't cover the
specific case of guitar strings and pickups (damn!), but it does
develop equations for vibrating strings from an analysis of
standing waves. A slight variation on one of the textbook
equations provides the output of a pickup at positions along a
string:

Where:

Vpickup is the relative
displacement velocity, and thus the relative pickup
output level, at this point on the string. A value
of 1.0 is the maximum.

Xpickup is the position of the pickup,
the distance between the bridge and the center of the pickup in
inches.

Lvib is the vibrating length of
the string in inches. For a harmonic, use the
distance from the bridge to the first node. For the
fundamental mode on an open string, this is the scale length.

Throughout this article I specify linear distances in inches
because most guitar specifications are in inches. In every
equation the linear units are used in ratios, so any consistant
linear unit will work fine.

A frequency response can be computed by relating the vibrating
string length to frequency. It's an inverse relationship.

Where:

Fstring is the frequency of the vibrating
string in Hz, whether the pitch is due to the fundamental or a
harmonic of a fretted note or open string.

Fopen is the fundamental open (unfretted)
frequency of the string in Hz.

Lscale is the length of the open (unfretted)
string in inches.

Substituting in:

Figure 4 is a plot of this function on a linear frequency
horizontal scale and dB amplitude vertical scale . There
are null points in the response every Fnode Hz, where
Fnode is:

Fnode is also the pitch of the string if the
string was fretted at the pickup location.

This curve is called a "comb filter" because the frequency
response shape resembles the teeth of a comb. Comb filter
responses are often associated with time delays (echoes, flange
effects, chorus effects). Here that time delay is related to
waves on the string reflecting off the bridge and the fret or nut.

Figure 5 shows the frequency response on a log-log plot (log
frequency and dB amplitude) of a neck position pickup as on a
Stratocaster low E string (82 Hz open string, 25.5 inches scale,
pickup located 6.375 inches from bridge).

Figure 5. Response of the neck pickup on the low E
string, 82 Hz, 25.5-inch scale, pickup located 6.375 inches
from the bridge. The first peak is at 165 Hz, the
first null is at 330 Hz.

(As an aside, the first time I plotted this curve was in high
school, probably around 1973. That was a BASIC program
running on a Hewlett-Packard 2114B minicomputer, outputting dots
to an ASR 33 Teletype. The version you see here is a three
page PostScript program. This version is much prettier.)

As a reference, the colored bar above the curve shows the
frequency range of the fretted notes on a two-octave
fingerboard. Frequencies to the right of the bar are thus
only available as harmonics, or if the instrument is played with a
slide. Guitar strings cannot support subharmonics, so
sustained frequencies to the left of the bar would only be
audible if the neck length were extended.

This comb filter effect means that the notches will be cramped
together at the high end of the log frequency plot. The
chaotic nature of the response plot at the high end is due to
drawing limitations; I've plotted the curve at each horizontal
pixel and at the far right each pixel covers a number of comb
filter cycles. Note that most pickups cut off
frequencies above 8 kHz or so depending on the inductance and
capacitance of the pickup.

Figure 6 shows how the response curve shifts for the neck,
middle, and bridge pickup positions.

Note that the low end of the response drops off as the pickup
moves toward the brige. The output of the open string is
roughly 10dB lower on the brige pickup compared to the neck
pickup. And note how the frequencies of the first peak
rise as the pickup moves toward the bridge.

On an electric instrument these comb filter curves provide an
effect functionally similiar to the body resonances of an
acoustic guitar, providing an interesting character to the
sound. (The actual body resonances of an electric
instrument are certainly significant to electric sound, but
not in the same way as the acoustic effect).

One aspect about these curves that is very different from the
body resonances of acoustic instruments is that the curve is
with respect to the string tuning, and the curve will scale up
and down with the pitch of the open (non fretted)
string. Figure 7 shows a set of plots similiar to Figure
6, but for the high E string tuned to 328 Hz.

Neck pickup, 6.375 inches from the bridge, first peak
is at 659 Hz, the first null is at 1319 Hz.

And since we're talking about moving pickups around, this
would be a good time to point out that there have been at
least two instruments in production with sliding pickups;
the Gibson "Grabber" bass and a Dan Armstrong model.

Here is a drawing from Stanley Rendell's Gibson patent
US 3,911,777: Electric guitar with slidable pickup beneath
strings. You can check out the patent
below.

Effects of pickup width

Pickups do not sense the string at a single point source, but
rather over an area due to the width of the magnetic field.
This sensing area is called the "aperture" of the pickup and is
about an inch wide on a thin single coil pickup and about 2.5
inches wide on a wider pickup such as the Gibson humbucker.

The effect of the pickup aperture on the response can be
calculated by averaging the point response over the aperture
length. This isn't completely accurate, the pickup
sensitivity will be greater in the middle than at the ends,
but this makes a fine first approximation.

Where:

Wwidepickup is the width of the wide pickup in inches.

Performing the integration:

Given the classic geometric equality from high school math:

And its companion:

Subtracting the first from the second:

Applying this to our equation above:

Figure 10 shows the response of a neck pickup 1.0 inches wide on
the low E of a Stratocaster. Figure 11 is the same thing,
but the pickup is 2.5 inches wide. Note that the original
point pickup response is readily apparent, but it is multiplied
by a second comb filter.

This form of the equation is especially interesting because the
two parts of the equation are separable -- the comb filter due
to the pickup position is separate from the comb filter due to
the pickup width. Here is the equation for the effect of
pickup width alone:

Where:

Vpickupwidth is the relative output
voltage level due to the pickup width only.

Figures 12 and 13 are response plots of the only effect of
pickup width of pickups 1.0 inch and 2.5 inches wide
respectively. This is a 6dB per octave low-pass comb
filter with a -3dB tuning of approximately 0.866
LscaleFopen/Wpickup.
The effect of pickup width scales with the pitch of the
open string, but is independant of pickup position. Each
of the nulls is due to an integral number of wavelengths under
the pickup.

Also interesting to note is that this low pass filter effect is
without phase shift or delay from the original string vibration.

Roland fakes it

Roland has a device out
called the VG-8 Virtual Guitar System that attempts to
simulate the effects of various pickup placements with digital
filters. I have not played this unit, so I can't comment
on its performance. But since it implements the response
curves above and the patents for it are readily available, I
thought it would be good to include them here. The
drawings here are from the patents.

Atsushi Hoshiai's US Patent 5,367,120, Musical tone signal
forming device for a stringed musical Instrument
shows most of the operation. First the basic setup; a
hex pickup attaches to an electric guitar and connects to the
main circuitry. A hex pickup is necessary because the
filtering for each string will be different and dependant
on the tuning of each string.

The comb filter is implemented by subtracting a delayed
version of the signal from the incoming signal for that
string. This produces a comb filter response simliar to
those I have plotted above. A pitch detection scheme is
used to be able to set the delay relative to the string's open
frequency.

The hex pickup is placed close to the bridge. The
response of the hex pickup is flatter there, but it will still
contain some nodes and antinodes. An inverse comb filter
is used compensate for this effect.

Finally a method is presented for emulating the response of a
mix of two pickups in different positions using a series
connection of an average pickup position comb filter and a
pickup separation comb filter. I cover the math behind
this in my article
Response Effects of Guitar Pickup Mixing.
A better approach to emulating multiple pickups is described
in the next patent.

Atsushi Hoshiai's US Patent 5,731,533: Musical tone signal
forming apparatus for use in simulating a tone of string
instrument covers the case of simulating two or more
pickups. This patent is written in an incredibly
confusing manner and it took me a while to figure out what
it was all about. I think I can summarize it better:

While simply subtracting a delayed version of the input signal
will generate a comb filter response similar to a single pickup
a given distance from the bridge, it doesn't work to extend this
to multiple pickups because the time delay relationships between
the pickups will not be correct. What's required is to
center the delays relative to a point one half the longest
delay. That's it.

Conclusion

A physical analysis provides equations for the frequency
response of an electric guitar pickup due to the physical
position and magnetic aperture of the pickup. Response
plots are drawn from these equations and compares well with
musician's experience. The pickup position response is a
comb filter curve and the pickup width response is a 6dB per
octave low pass comb filter response.

These equations only apply to the mechanical effects of position
and width for sustained fundamental and harmonic
vibrations. Certainly there are many more factors that
affect the sound, including body resonances, the initial
transient, electrical characterists of the pickups, and so
forth.

Footnotes

Fender Stratocaster specs

Throughout this article I use the Fender Stratocaster as an
example because it is a remarkably popular instrument, probably
the most popular guitar model over the last 30 years, and because
it has three pickups. I've also owned one for over ten years
and have a good feel for the instrument.

These are not definitive manufacturing specs for the
Stratocaster. The pickup position numbers are measured off
of my personal guitar, the neck pickup position is adjusted 0.125
inch to be exactly 1/4 the scale length and I'm ignoring the fact
that the bridge pickup is mounted on a slant and just using an
average position for it.

I'm also ignoring bridge compensation.

Fender Jazz Bass specs

I also use the Fender Jazz Bass for the bass plots. One of
my favorite instruments for sure, and a very popular bass model.